A082540 -id:A082540 - cp's OEIS frontend

A344038 Number of ordered 6-tuples (a,b,c,d,e,f) with gcd(a,b,c,d,e,f)=1 (1<= {a,b,c,d,e,f} <= 10^n).

1, 983583, 983029267047, 982960635742968103, 982953384128772770413831, 982952672223441253533233827367, 982952600027678075050509511271466303, 982952593055042000417993486008754893529583, 982952592342881094406730790044111038427637071855

Offset: 0

Karl-Heinz Hofmann, May 07 2021

Chai Wah Wu, Table of n, a(n) for n = 0..15

Cf. A342586, A342841, A343193, A343282, A343359, A013664.
Cf. A343978, A071778, A018805, A082540, A082544.
Related counts of k-tuples:
pairs: A018805, A342632, A342586;
triples: A071778, A342935, A342841;
quadruples: A082540, A343527, A343193;
5-tuples: A343282;
6-tuples: A343978, A344038. - N. J. A. Sloane, Jun 13 2021

a(n)={sum(k=1, 10^n+1, moebius(k)*(10^n\k)^6)} \\ Andrew Howroyd, May 08 2021

from labmath import mobius
def A344038(n): return sum(mobius(k)*(10**n//k)**6 for k in range(1, 10**n+1))

Lim_{n->infinity} a(n)/10^(6*n) = 1/zeta(6) = A343359 = 945/Pi^4.

a(n) = A343978(10^n).

Edited by N. J. A. Sloane, Jun 13 2021

A332468 a(n) = Sum_{k=1..n} mu(k) * floor(n/k)^n.

Original entry on oeis.org

1, 3, 25, 239, 3091, 45863, 821227, 16711423, 387138661, 9990174303, 285262663291, 8913906888703, 302861978789371, 11111328334033327, 437889112287422401, 18446462446101903615, 827238009323454485641, 39346257879101283645743, 1978418304199236175597105

Offset: 1

Ilya Gutkovskiy, Feb 13 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Main diagonal of A344527.

Cf. A008683, A018805, A071778, A082540, A082544, A332469, A343978.

[&+[MoebiusMu(k)*Floor(n/k)^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 13 2020

Table[Sum[MoebiusMu[k] Floor[n/k]^n, {k, 1, n}], {n, 1, 19}]
b[n_, k_] := b[n, k] = n^k - Sum[b[Floor[n/j], k], {j, 2, n}]; a[n_] := b[n, n]; Table[a[n], {n, 1, 19}]

a(n)={sum(k=1, n, moebius(k) * floor(n/k)^n)} \\ Andrew Howroyd, Feb 13 2020

from functools import lru_cache
@lru_cache(maxsize=None)
def A344527_T(n,k):
    if n == 0:
        return 0
    c, j, k1 = 1, 2, n//2
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A344527_T(k1,k)
        j, k1 = j2, n//j2
    return n*(n**(k-1)-1)-c+j
def A332468(n): return A344527_T(n,n) # Chai Wah Wu, Nov 02 2023

a(n) ~ n^n. - Vaclav Kotesovec, May 28 2021

A344597 a(n) = Sum_{k=1..n} mu(k) * (floor(n/k)^4 - floor((n-1)/k)^4).

Original entry on oeis.org

1, 14, 64, 160, 368, 592, 1104, 1520, 2400, 3056, 4640, 5264, 7824, 8736, 11776, 13216, 17984, 18384, 25344, 26080, 33312, 35120, 45584, 44320, 58480, 58512, 72000, 73200, 92624, 86848, 113520, 110144, 132640, 132416, 162816, 152112, 194544, 185616, 220416

Offset: 1

Seiichi Manyama, May 24 2021

nonn

Cf. A000583, A082540, A140434, A344596.

a[n_] := Sum[MoebiusMu[k] * First @ Differences @ (Quotient[{n - 1, n}, k]^4), {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 24 2021 *)

a(n) = sum(k=1, n, moebius(k)*((n\k)^4-((n-1)\k)^4));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^4))

from functools import lru_cache
@lru_cache(maxsize=None)
def A082540(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A082540(k1)
        j, k1 = j2, n//j2
    return n*(n**3-1)-c+j
def A344597(n): return A082540(n)-A082540(n-1) # Chai Wah Wu, May 09 2025

Sum_{k=1..n} a(k) * floor(n/k) = n^4.
Sum_{k=1..n} a(k) = A082540(n).
G.f.: Sum_{k >= 1} mu(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^4.

A101467 Number of distinct n-term ratios x_1 : x_2 : ... : x_n where each x_i is in the range [1-10].

Original entry on oeis.org

10, 63, 841, 9279, 96601, 983583, 9919561, 99602559, 998026681, 9990174303, 99950992681, 999755323839, 9998777694361, 99993891685023, 999969468040201, 9999847368997119, 99999236931275641, 999996184915051743, 9999980925350886121, 99999904629080526399

Offset: 1

Su Jianning (sujianning(AT)yahoo.com.cn), Jan 21 2005

Number of elements of {1,...,10}^n with gcd 1. - Robert Israel, Nov 28 2014

			For n=2: Consider the ratios 1:1, 1:2, ..., 1:10, 2:1, 2:2, ..., 2:10, ..., 10:1, 10:2, ..., 10:10. We get 63 different ratios from the 100 numbers list above after removing duplication. So a(2) = 63, and this is A018805(10).

Colin Barker, Table of n, a(n) for n = 1..999
Index entries for linear recurrences with constant coefficients, signature (21,-151,471,-640,300).

Cf. A018805 (2 terms), A071778 (3 terms), A082540 (4 terms), A082544 (5 terms).

1, seq(10^n - 5^n - 3^n - 2^n + 1, n=2..20); # Robert Israel, Nov 28 2014

Vec(x*(2700*x^5-5460*x^4+3579*x^3-1028*x^2+147*x-10)/((x-1)*(2*x-1)*(3*x-1)*(5*x-1)*(10*x-1)) + O(x^100)) \\ Colin Barker, Nov 28 2014

a(1) = 10; for n>1, a(n) = 10^n - 5^n - 3^n - 2^n + 1.
G.f.: x*(2700*x^5-5460*x^4+3579*x^3-1028*x^2+147*x-10) / ((x-1)*(2*x-1)*(3*x-1)*(5*x-1)*(10*x-1)). - Colin Barker, Nov 28 2014
a(n+4) = -300*a(n)+340*a(n+1)-131*a(n+2)+20*a(n+3)+72 for n >= 2. - Robert Israel, Dec 02 2014
a(n) = 21*a(n-1) - 151*a(n-2) + 471*a(n-3) - 640*a(n-4) + 300*a(n-5) for n > 6. - Chai Wah Wu, Apr 15 2021

A383783 a(n) = Sum_{k=1..2^n} mu(k) * (floor(2^n/k)^4 - floor((2^n-1)/k)^4).

Original entry on oeis.org

1, 14, 160, 1520, 13216, 110144, 899200, 7266560, 58425856, 468583424, 3753379840, 30045900800, 240442679296, 1923843375104, 15391954862080, 123140470538240, 985143091265536, 7881222038749184, 63050085546065920, 504401921315962880, 4035220318323736576

Offset: 0

Chai Wah Wu, May 09 2025

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..48

Cf. A082540, A344597.

a[n_]:=Sum[MoebiusMu[k]*(Floor[2^n/k]^4-Floor[(2^n-1)/k]^4),{k,2^n}]; Array[a,21,0] (* James C. McMahon, May 10 2025 *)

a(n) = sum(k=1, 2^n, moebius(k) * ((2^n\k)^4 - ((2^n-1)\k)^4)); \\ Michel Marcus, May 10 2025

from functools import lru_cache
@lru_cache(maxsize=None)
def A082540(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A082540(k1)
        j, k1 = j2, n//j2
    return n*(n**3-1)-c+j
def A383783(n): return A082540(m:=1<A082540(m-1)

a(n) = A344597(2^n) = A082540(2^n) - A082540(2^n-1).

cp's OEIS Frontend

A344038 Number of ordered 6-tuples (a,b,c,d,e,f) with gcd(a,b,c,d,e,f)=1 (1<= {a,b,c,d,e,f} <= 10^n).

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A332468 a(n) = Sum_{k=1..n} mu(k) * floor(n/k)^n.

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A344597 a(n) = Sum_{k=1..n} mu(k) * (floor(n/k)^4 - floor((n-1)/k)^4).

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A101467 Number of distinct n-term ratios x_1 : x_2 : ... : x_n where each x_i is in the range [1-10].

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A383783 a(n) = Sum_{k=1..2^n} mu(k) * (floor(2^n/k)^4 - floor((2^n-1)/k)^4).

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